Nonmonotonic Logics and Semantics

Daniel Lehmann
Hebrew University, Jerusalem, Israel

Abstract

Tarski gave a general semantics for deductive reasoning: a formula a may be deduced from a set A of formulas iff a holds in all models in which all the elements of A hold. A more liberal semantics has been considered: a formula a may be deduced from a set A of formulas iff a holds in all of the "preferred" models in which all the elements of A hold, where the notion of "preferred" satisfies a set of natural properties. This semantics is here shown to be equivalent to a semantics based on comparing the relative "importance" of sets of models, by what amounts to a qualitative probability measure. The consequence operations defined by the equivalent semantics are then characterized by a weakening of Tarski's properties in which the monotonicity requirement is replaced by three weaker conditions. Classical propositional connectives are characterized by natural introduction-elimination rules in a nonmonotonic setting. Even in the nonmonotonic setting, one obtains classical propositional logic.
Eyal Amir
Last modified: Wed Sep 9 11:18:55 PDT 1998